where \(\Phi (\cdot )\)  denotes the cumulative distribution function (cdf)
of a standard normal distribution and \(I(y_{i}=j)\)  is an indicator function,
which equals 1 if the condition within parenthesis is true and 0 otherwise.
The parameter estimates for (\(\beta , \gamma \) ) are obtained by maximizing the
logarithm of the likelihood (equation REF ) using numerical
techniques such as the Newton-Raphson method or BHHH procedure
[1]}. The principle behind maximizing the likelihood – known
as maximum likelihood (ML) estimation – is to obtain those parameter values
that are most probable to have produced the data under the assumed
statistical model. Note that it is convenient to work with the logarithm of
the likelihood (log-likelihood) since logarithm being a monotonic function,
the maximum of the log-likelihood and the likelihood occur at the same
parameter values. Once the parameter estimates are available, they may be
used to calculate the covariate effects, make predictions or assess model
fitting. Interested readers may look into [2]} or
[3]} for a detailed review of ordinal data
modeling.
